Filippov's Solution and Finite-Time Stability of Stochastic Systems for Discontinuous Control

In this article, a suite of theoretic tools is provided for discontinuous control design and finite-time stability analysis of a class of stochastic differential systems. The notion of Filippov's solutions for stochastic differential systems is proposed, and the corresponding solution existence problem is explored. The classical Itô differentiation formula is generalized for quasi-$C_{0}^{2}(\mathbb {R}^{n},\mathbb {R})$-class functions along Filippov's solutions of stochastic differential systems, and two involved set-valued stochastic integrals are introduced with a study on their properties. Some finite-time stability results of stochastic differential systems are revealed with Filippov's solutions, and one of them is applied to neural synchronization, together with case simulations.